MASERS, INTERSTELLAR AND CIRCUMSTELLAR, THEORY

Moshe Elitzur

The radio radiation detected in some lines of certain astronomical
molecules is attributed to the natural occurrence of the maser
phenomenon (microwave amplification by stimulated emission of
radiation), the same as that produced by artificial means in laser
devices. Astronomical maser radiation is produced by population
inversion of the pertinent transitions and is usually identified as
such when at least one of the following properties is observed:

Enormous radiation intensity, as measured by its equivalent temperature.

Very narrow linewidths.

Abnormal line ratios, indicating deviations from equilibrium.

In at least one astronomical maser source there is also direct
evidence for population inversion and amplification of radiation. The
mechanism that leads to inversion, which is caused by the cycling of
molecules through energy levels, is called the pump. The pump process
is initiated by excitations from the ground state, due to either
external radiation or collisions. The inversion results from a
combination of various factors, specific to the particular maser
molecule. The strong maser emission detected in our galaxy occurs
around late-type stars, where it is called circumstellar, and in the
cores of dense molecular clouds that are regions of active star
formation, which is then termed interstellar. Maser radiation probes
small-scale structure in these sources and is now used to measure
distances by kinematic means, the equivalent of the classic moving
cluster method. In recent years, maser emission has also been detected
in many nearby galaxies.

MASER EFFECT

Population exchange between the two levels of any transition is
governed by both collisional and radiative processes. The latter
include spontaneous decays from upper to lower level and absorption of
external radiation, accompanied by an excitation from lower to upper
level. The frequency of the absorbed photon must match the energy
separation of the transition. The inverse process, induced or
stimulated emission, is the essence of the maser phenomenon. In it, a
downward transition is induced by an incoming photon with a matched
frequency. To conserve energy and momentum, the transition is
accompanied by the emission of another photon whose properties are
identical to those of the initial parent photon. The process
effectively acts as negative absorption - increasing, instead of
decrease the number of photons in the radiation field. If for any
reason the population density of the upper level is larger than that
of the lower level, the rate of stimulated emission exceeds absorption
and the medium amplifies the propagating radiation rather than
attenuating it. When the contribution of stimulated emission is
included in the absorption coefficient, the latter becomes
negative. The same applies to the optical depth and the standard
attenuation term exp(-) becomes
an amplification factor exp||. The
absolute value of the optical depth is then referred to as the maser
gain. A gain of more than 20 leads to amplification in excess of
108 and could explain in part the observed exceptional
intensities. Additional intensity enhancement results from the
focusing of the radiation into a narrow beam, caused by the fact that
amplification is proportional to incoming intensity, so stronger rays
are amplified more strongly. Observed brightness is frequently
expressed in terms of brightness temperature, the temperature of an
equivalent blackbody that would be needed to produce the same
intensity. Maser brightness temperatures can be as high as ~
1015 K.

The exponential growth of the intensity cannot continue indefinitely
because it would eventually lead to infinite energy density for
sufficiently long masers. A self-limiting process is built right into
the maser effect itself: Induced emission removes particles from the
upper level, thus reducing the inversion. Because of the excess of
induced emissions over absorptions, the inversion decreases once the
population exchange between the maser levels is dominated by the
interaction with the maser radiation. The intensity then approaches a
limit and the maser saturates. The brightness is highest during
saturated operation because every pumping event produces a maser
photon with the maximal intrinsic efficiency allowed by the pump.

ENVIRONMENTAL CONSTRAINTS

The prevalence of astronomical masers shows that the interstellar
medium is apparently capable of producing the maser effect relatively
easily, although special efforts are required to achieve the same end
in the laboratory. This is a result of the great differences in
densities and geometrical dimensions between the two environments. In
thermodynamic equilibrium, the populations per substate n of the two
levels of a transition separated by energy gap E obey.

nupper / nlower = exp (-E / KT)

where k is the Boltzmann constant and T is the temperature. This
equilibrium distribution is established when population exchange is
dominated by collisions, which is the case for sufficiently high
densities. For typical parameters of molecular rotational transitions,
the required densities are in excess of ~ 104
cm-3. This is an
extremely low density in the laboratory, achieved only with state of
the art vacuum techniques (pressures of ~
10-12-10-13 torr), but it is
quite high by interstellar standards. Equilibrium populations are
therefore the rule in terrestrial circumstances but are the exception
in interstellar space due to the large difference in relevant
densities. Once thermal equilibrium is violated, population inversion
is a priori almost as likely as its reverse.

An appreciable maser effect requires large gain, which in turn
implies a substantial number of molecules along the line-of-sight.
This conflicts with the necessary deviation from thermal equilibrium,
which requires that the densities be small, and the only way to
reconcile these opposite demands is with large dimensions. At a
density of ~ 104 cm-3, an appreciable gain is
achieved only for linear
dimensions in excess of ~ 1010 cm - almost as much as the
radius of the
Sun. One of the methods to overcome this difficulty in the laboratory
is to increase the radiation path length by bouncing the laser light
between mirrors, effectively increasing the linear dimensions of the
system by the many passes of the laser beam in the resonant
cavity. This technique is possible only in systems that can maintain a
high degree of phase coherence. On the other hand, typical lengths of
astronomical masers are at least ~ 1013 cm (the same as the
radius of
the Earth's orbit around the Sun), and the required gains are obtained
during simple photon propagation as in a single-pass
laser. Interstellar space is therefore a natural environment for maser
operation. Using laboratory terminology, astronomical masers are
single-pass, lossless, gaseous lasers without feedback.

The observed profiles of interstellar lines usually indicate the
presence of highly supersonic motions in the emitting regions. Because
maser amplification is achieved by induced emission, the maser photons
must seek paths that maintain good coherence in the component of the
velocity along the line-of-sight; otherwise the transition frequencies
of molecules encountered by the maser photons would be shifted by the
Doppler effect, making amplification impossible. Observations show
indeed that maser sources are comprised of many emission spots, each
with its own well defined velocity. These single features are often
shaped like elongated tubes, or cylinders, which need not be
well-defined physical entities, but rather directions that developed
the required velocity coherence by chance. Because of their high
brightness, the spots can be studied individually using techniques of
high-resolution interferometry. This provides the opportunity for
probing small scale structure in the host environments.

SOURCES AND PUMPING

Maser emission is associated with both the early and late stages in
the life of a star. This is a fortunate coincidence because these are
generally regarded as the most interesting phases of stellar
evolution. Late-type stars display strong maser radiation in
transitions of all three "classical" maser molecules - OH, H2O, and
SiO. The masers occur in distinct regions located at different
distances from the central star. The SiO masers involve rotational
transitions inside excited vibration states, which lie high above the
ground state. These levels can maintain substantial populations only
close to the star where the excitation rates are high, and that is
where SiO masers are located. The H2O and OH masers, on the other
hand, emanate from transitions in the ground vibration states of these
molecules and do not require such extreme conditions for pumping. Both
are located in shells that are part of an expanding wind that blows
away from the star. The H2O masers require higher temperatures
than OH masers and occur at distances of up to ~ 1015 cm from
the central star
(by comparison, the radius of Pluto's orbit around the Sun is about
6 x 1014 cm). The OH maser shell extends further out, to a
radius larger by another order of magnitude at least.

Regions of active star formation, located at the cores of many
molecular clouds, display the most powerful and spectacular maser
emission observed in the Galaxy in both OH and H2O, and in at least
one case also in SiO. The OH masers appear to be surface phenomena on
the edges of very compact H2O regions, ionized spheres around young
and very hot stars. The H2O masers usually trace high velocity flows
(velocities of ~ 200 km s-1) from some centers of activity that
presumably erupt at a certain stage of the star formation process.

Pump analysis involves the construction of models capable of
producing the observed maser output with parameters that can be
checked against other observations. The pumping mechanism is termed
collisional or radiative according to the nature of the
process that
dominates the molecular excitation from the ground state. In general,
both types of pumping can produce inversion under the right conditions
(although in certain, specific circumstances only one may be capable
of inversion) and the nature of the pump in any particular source is
determined by the relative strengths of both processes. Radiative pump
models can be more easily confronted with observations because they
relate two directly observed quantities - the number of photons observed
in the maser transition and in the pump bandwidth. Indeed, the masers
whose detailed modeling has been most successful are the OH masers in
late-type stars that are pumped by infrared radiation resulting from
the reemission of the stellar radiation by the dust particles that
permeate the stellar wind. Detailed models of the H2O masers
in these sources show that pumping is controlled by collisions. The model
predicts correctly the location of the H2O maser region and its
variation with the stellar mass loss rate, but these observational
tests are not as direct. Current models of masers in star-forming
regions are not as detailed yet, reflecting perhaps the somewhat
poorer overall understanding of these sources.